GR coordinate acceleration problem

Let's say we have a mass with an object orbitting with constant speed in a circular orbit and a distant observer Bob. According to Bob's coordinate system, the orbit is circular at a speed v and a constant inward coordinate acceleration a. The coordinate acceleration is just what is inferred according to Bob's coordinate system by the usual definitions of acceleration, s = vo t + 1/2 a t^2, 2 a s = vf^2 - vo^2, a = (vf - vo) / t, etc., where s is the distance travelled and vo and vf are the original and final velocities. The relativistic acceleration formulas also reduce to these when considering a body starting at rest and accelerating an infinitesimal distance over infinitesimal time, which we will be doing, so we need not worry with more complicated formulas for this.

So if the object is initially at coordinates x_o=r, y_o=0, and travels an infinitesimal distance y = v t in the tangent direction, it will have travelled a distance of x = r - sqrt(r^2 - y^2) in the radial direction toward the mass, since the orbit is perfectly circular. The distance being infinitesimal, we can drop higher orders and gain just x = r - r (1 - y^2 / (2 r^2)) = y^2 / (2 r) in the radial direction. We want the radial coordinate acceleration as inferred by Bob, so we can use s = vo t + 1/2 a t^2, where vo = 0 for the original velocity in the radial direction, so x = s = 1/2 a t^2.

So now we have x = 1/2 a t^2 = y^2 / (2 r), and since y = v t, this reduces to just

1/2 a t^2 = y^2 / (2 r)

a t^2 = (v^2 t^2) / r

a = v^2 / r

giving the usual acceleration formula for a circular orbit. Again, this is just the coordinate acceleration inferred by Bob's coordinate system. It is not saying anything about proper acceleration or anything else, just a coordinate effect. Does all of this look okay so far?

Your calculation looks right but your interpretation is not correct in the absence of gravity. In that case the acceleration is proper and the orbiting body will experience the force which keeps it in orbit.

Your calculation looks right but your interpretation is not correct in the absence of gravity. In that case the acceleration is proper and the orbiting body will experience the force which keeps it in orbit.

Okay good, thanks. I only want to measure the radial coordinate acceleration as inferred by Bob's coordinate system, independent of whatever the proper acceleration might be.

Okay, so next let's say we have a hovering observer Alice at r. At the moment the orbitting object passes Alice, Alice drops a ball from rest at r. We can say that the object and the ball coincide in the same place at that time. Now, according to the equivalence principle, the radial coordinate acceleration that Alice measures at that instant for both the object and the ball will be the same, correct? The object has no radial velocity, so it should be as if dropped from rest at that instant the same as the ball and accelerate radially with the ball for a moment, right? Although the object is also still travelling tangently.

As for the distant observer Bob, the same should be true according to his coordinate system, shouldn't it? He infers that the ball and the object both coincide at the same place at the same time and at that moment both coordinately accelerate radially from rest in the same way, right? Although of course, the radial coordinate acceleration Bob infers will be different from that which Alice measures locally.

Staff: Mentor

according to the equivalence principle, the radial coordinate acceleration that Alice measures at that instant for both the object and the ball will be the same, correct?

It depends on what spacetime you're in. If you're in flat spacetime, the radial coordinate acceleration of the dropped object will be zero in Alice's rest frame, because Alice will just be an inertial observer at rest in Bob's global inertial frame, and the dropped object is moving inertially and starts at rest (with Alice), so it just stays at rest. The ball, OTOH, if you're in flat spacetime, will have an inward coordinate acceleration of ##v^2 / r##, as you calculated, because it has a force being exerted on it to hold it in a circular path.

If, OTOH, you are in Schwarzschild spacetime, at a sufficiently large radius (see below for why), then yes, the radial coordinate acceleration of both the dropped object and the ball, in Alice's momentarily comoving inertial frame, will be the same, because Alice is not an inertial observer; she must be firing rockets or otherwise be subject to a force that accelerates her radially outward. Both the ball and the dropped object are moving inertially, so their radial coordinate acceleration will be the same.

(The reason why we have to be at a sufficiently large radius is that the ball's orbital velocity must be much less than the speed of light; if it's a significant fraction of the speed of light, then the ball's radial coordinate acceleration will be larger than that of the dropped object because of relativistic effects--the same effects that make the bending of light by the Sun twice as large as a naive Newtonian calculation would predict.)

the coordinate acceleration Bob infers will be different from that which Alice measures locally.

This also depends on the spacetime; it's true in Schwarzschild spacetime, but false in flat spacetime: in flat spacetime, since Alice and Bob are both inertial observers and are at rest relative to each other, all coordinate accelerations are the same for both of them.

If, OTOH, you are in Schwarzschild spacetime, at a sufficiently large radius (see below for why), then yes, the radial coordinate acceleration of both the dropped object and the ball, in Alice's momentarily comoving inertial frame, will be the same, because Alice is not an inertial observer; she must be firing rockets or otherwise be subject to a force that accelerates her radially outward. Both the ball and the dropped object are moving inertially, so their radial coordinate acceleration will be the same.

Right, this is what I am describing, although the ball is dropped and the object orbits :)

(The reason why we have to be at a sufficiently large radius is that the ball's orbital velocity must be much less than the speed of light; if it's a significant fraction of the speed of light, then the ball's radial coordinate acceleration will be larger than that of the dropped object because of relativistic effects--the same effects that make the bending of light by the Sun twice as large as a naive Newtonian calculation would predict.)

Okay, this is what I would be interested in. Before you said the coordinate acceleration at that instant when they coincide in the same place would be the same, but here it looks like something different. Let's throw in a relativitistic speed. I understand that the curvature of spacetime varies from Newtonian effects, giving twice the overall bending of light, but let's say for a moment that light passes tangently at the same moment that Carl falls from rest in an elevator at the same place as Alice. Wouldn't the coordinate acceleration as measured by Alice for both be the same at that moment? It would be the same as if Carl emitted light sideways in his elevator upon falling, would it not? So for a moment that light would continue to travel perpendicularly to Carl according to the equivalence principle, while both the elevator and the light coordinately accelerate radially at the same rate according to Alice, correct?

I will cut to the chase. The problem is I am still trying to find relationships and invariants that are independent of the coordinate system used, as we did here. There we found the relations

a' = G M L_t^2 / (z r^2) and a' = L z' c^2 / z

where at r, a' is the locally measured acceleration, z is the time dilation, L is the radial length contraction, and L_t is the length contraction in the tangent direction. From the derivatives of these two relations we can gain the R_00 tensor.

Okay, so now I want to look at what the distant observer infers and find relations and invariants from that. So far we have

a = v_t^2 / r

where v_t is the tangent velocity. From the equivalence principle, if the hovering observer Alice were at r in the same place as the object passes and Alice lets go of a ball at that moment, the ball and object should both accelerate toward the mass at the same rate at that instant. Although the object is travelling tangently, its initial radial velocity is zero, the same as the ball.

Alice measures the local acceleration to be a'. She measures the ball to fall some distance radially along the length of a short rod over some time. The distant observer Bob, however, infers that the time that passes for the ball to travel the length of the rod is 1/z longer and that the length of the rod itself is L shorter, so the coordinate acceleration that Bob measures is z^2 L smaller, whereby a = z^2 L a'. Likewise, the locally measured tangent velocity is v_t', and Bob infers 1/z greater time to travel a distance that is L_t shorter in the tangent direction, so v_t = z L_t v_t'. So now we have

z^2 L a' = (z L_t v_t')^2 / r

L a' = L_t^2 v_t'^2 / r

Here's where the problem lies. Let's look at the invariants. L_t / r is an invariant, having the same value in any coordinate system. a' and v_t' are also invariants since they are what is measured locally, regardless of the distant observer's coordinate system. But that leaves

L / L_t = (L_t / r) v_t'^2 / a'

The right side is all invariant, so the left side should be too. But if the ratio of the radial to tangent length contraction is invariant, then there can be one and only one valid coordinate system. For instance, if one coordinate system gives some ratio L / L_t as an invariant for some spherical shell (with L being found between two very close shells), then the only way to change the coordinate system such that this ratio remains the same for that shell is to change the radius of all shells by the same ratio. But if we make them all .99 the original radiuses, then the distant observer's distance from the mass also changes by .99, whereas it should verge upon 1, so there can be only one coordinate system where this is also true.

Let's go ahead and try this for the Schwarzschild coordinate system and see if we can pinpoint where the problem actually lies. What is the locally measured orbittal speed as measured by a hovering observer at r for an object in a circular orbit at r?

Let's go ahead and try this for the Schwarzschild coordinate system and see if we can pinpoint where the problem actually lies. What is the locally measured orbital speed as measured by a hovering observer at r for an object in a circular orbit at r?

When the two worldlines coincide they will measure a relative velocity of ##\sqrt{\frac{m}{r-2\,m}}## which is close to ## \sqrt{\frac{m}{r}}## if ##r>>m##.

This comes from ##-\gamma = V^aU_a ## where ##U## and ##V## are the respective worldlines.

Oh, z is also an invariant, so we are left with L_t^3. It will work out, however, if v_t^2 = L_t^3 G M / r, although I don't immediately see why that should be the case. Plugging that into the coordinate acceleration formula for the distant observer, we get

a = v_t^2 / r

a = G M L_t^3 / r^2

z^2 L a' = G M L_t^3 / r^2

and separating the invariants,

L / L_t = G M (L_t / r)^2 / z^2

so we are still left with L / L_t as an invariant. It would be an easy fix if a = z^2 L_t a' instead of a = z^2 L a', but the acceleration is in the radial direction, not the tangent direction.

Let's go ahead and try this for the Schwarzschild coordinate system and see if we can pinpoint where the problem actually lies. What is the locally measured orbittal speed as measured by a hovering observer at r for an object in a circular orbit at r?

I'm not following what you did, but I did something similar but more general in https://www.physicsforums.com/showthread.php?t=686147#post4354272
where I calculated a bunch of things for various general spherically symmetric metrics, one of which was the orbital velocity as measured by a co-located static observer. And of course the Schwarzschild metric is spherically symmetric.

Post #12 is the corrected version with all of the known typos removed (but I might have missed a few anyway).

Applying the results from that post:

The orbital velocity measured by a co-located static observer should be: